Transformations and Symmetry

A transformation is an operation that moves or changes a shape in some way to produce a new shape. Common types include translations, rotations, reflections, and dilations.

Translation is a type of transformation that slides a shape from one position to another without turning it. It is defined by a vector that indicates the direction and distance of the slide.

The rule for translating a point (x, y) by (a, b) is (x, y) --> (x + a, y + b).

Dilation is a transformation that changes the size of a shape but not its shape. It is defined by a scale factor.

The scale factor is a number that scales, or multiplies, the dimensions of a shape. A scale factor greater than 1 enlarges the shape, while a scale factor between 0 and 1 reduces it.

Two triangles are similar if their corresponding angles are equal and the lengths of their corresponding sides are proportional.

Rotation is a transformation that turns a shape around a fixed point called the center of rotation. The angle of rotation determines how far the shape is turned.

Reflection is a transformation that flips a shape over a line, creating a mirror image. The line is called the line of reflection.

Symmetry in geometry refers to a balance or similarity between two halves of a shape. A shape is symmetric if it can be divided into two identical parts.

Rigid transformations preserve the shape and size of a figure (e.g., translations, rotations, reflections), while non-rigid transformations change the size or shape (e.g., dilations).